Outline Formulation of the . . . Formulation of the . . . Are There Any Other . . . Why Decimal System and Considering the First . . . Binary System Are the Most Considering the First . . . How Do We Check the . . . Widely Used: A Possible How to Check the . . . The Checking and Its . . . Explanation Home Page Title Page Gerardo Muela Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, TX 79968, USA gdmuela@miners.utep.edu Page 1 of 11 Go Back Full Screen Close Quit
Outline Formulation of the . . . 1. Outline Formulation of the . . . • What is so special about numbers 10 and 2 that deci- Are There Any Other . . . mal and binary systems are the most widely used? Considering the First . . . Considering the First . . . • One interesting fact about 10 is that: How Do We Check the . . . – when we start with a unit interval and we want How to Check the . . . half-width, then this width is exactly 5/10; The Checking and Its . . . – when we want to find a square of half area, its sides Home Page are almost exactly 7/10, and Title Page – when we want to construct a cube of half volume ◭◭ ◮◮ its sides are almost exactly 8/10. ◭ ◮ • We show that b = 2, 4, and 10 are the only numbers Page 2 of 11 with this property – at least when b ≤ 10 9 . Go Back • This may be a possible explanation of why decimal and binary systems are the most widely used. Full Screen Close Quit
Outline Formulation of the . . . 2. Formulation of the Problem Formulation of the . . . • What is so special about numbers 10 and 2 that deci- Are There Any Other . . . mal and binary systems are the most widely used? Considering the First . . . Considering the First . . . • This questions was raised, e.g., by Donald Knuth in his How Do We Check the . . . famous book Art of Computer Programming . How to Check the . . . • One interesting fact about 10 is the following; when: The Checking and Its . . . – we start with a unit interval and Home Page – we want to constrict an interval of half width, Title Page – then this width is exactly 1/2 = 5/10. ◭◭ ◮◮ • When: ◭ ◮ – we start with a unit square and Page 3 of 11 – we want to find a square of area 1/2, Go Back � – its sides are 1 / 2, which is almost exactly 7/10: Full Screen � � � 1 2 − 7 1 � � � < 100 . � � Close 10 � � � Quit
Outline Formulation of the . . . 3. Formulation of the Problem (cont-d) Formulation of the . . . • Similarly, when: Are There Any Other . . . Considering the First . . . – we start with a unit cube and Considering the First . . . – we want to find a cube of volume 1/2, How Do We Check the . . . � – its sides are 3 1 / 2, which is almost exactly 8/10: How to Check the . . . � � The Checking and Its . . . � 2 − 8 1 1 � � 3 � < 100 . � � Home Page 10 � � � Title Page • So, whether we want to construct: ◭◭ ◮◮ – a piece of land which is (almost) exactly of half- ◭ ◮ area, or Page 4 of 11 – a piece of gold which is (almost) exactly of half- Go Back volume, Full Screen – decimal systems is very convenient. Close Quit
Outline Formulation of the . . . 4. Are There Any Other Numbers with This Formulation of the . . . Property? Are There Any Other . . . • Maybe here are other bases b with this property, i.e., Considering the First . . . for which, for some n 1 , n 2 , and n 3 , we have Considering the First . . . � � � � � � How Do We Check the . . . � � 2 − n 1 1 � < 1 2 − n 2 1 � < 1 2 − n 3 1 � < 1 � � � � � � 3 b 2 , b 2 , b 2 . � � � � � � How to Check the . . . b b b � � � � � � � The Checking and Its . . . • We show that – at least for b ≤ 10 9 – only the numbers Home Page b = 2, b = 4, and b = 10 satisfy this property. Title Page • Base 4 is, in effect, the same as the binary system: ◭◭ ◮◮ – we group 2 binary digits (bits) to get a 4-ary digit, ◭ ◮ – just like we get an 8-ary system when we Page 5 of 11 group 3 bits, or Go Back – we get a 16-based system when we group 4 bits. Full Screen • The above result may be a good explanation of why decimal and binary systems are the most widely used. Close Quit
Outline Formulation of the . . . 5. Considering the First Condition Formulation of the . . . • Let us first consider the first of the desired inequalities: Are There Any Other . . . � � 2 − n 1 1 � < 1 Considering the First . . . � � b 2 . � � b Considering the First . . . � How Do We Check the . . . • When the base is even, i.e., when b = 2 k for some How to Check the . . . integer k , then this property is clearly satisfied. The Checking and Its . . . • Indeed, in this case, for n 1 = k , we get n 1 b = 1 2 and Home Page � � 1 2 − k � = 0 < 1 Title Page � � thus, b 2 . � � b � ◭◭ ◮◮ • On the other hand: ◭ ◮ – if b is odd, i.e., if b = 2 k + 1 for some natural Page 6 of 11 number k ≥ 1, – then, for 1 2 = k + 0 . 5 2 k + 1 = k + 0 . 5 Go Back , the closest frac- b Full Screen tions of the type n 1 b are the fractions k b and k + 1 . b Close Quit
Outline Formulation of the . . . 6. Considering the First Condition Formulation of the . . . • For both fractions k b and k + 1 Are There Any Other . . . , we have b Considering the First . . . � k + 0 . 5 k � � k + 0 . 5 2 k + 1 − k + 1 � Considering the First . . . � � � � 2 k + 1 − � = � = � � � � 2 k + 1 2 k + 1 How Do We Check the . . . � � How to Check the . . . 0 . 5 2 · (2 k + 1) = 1 1 2 k + 1 = 2 b. The Checking and Its . . . Home Page • The desired inequality thus takes the form 1 2 b < 1 b 2 , Title Page which is equivalent to 2 b > b 2 and 2 > b . ◭◭ ◮◮ • However, odd bases start with b = 3. ◭ ◮ • So, the 1st condition cannot be satisfied by odd bases b . Page 7 of 11 Go Back • Thus, the first condition is equivalent to requiring that the base b is an even number. Full Screen Close Quit
Outline Formulation of the . . . 7. How Do We Check the Second Condition Formulation of the . . . � � � 2 − n 2 1 � < 1 � � Are There Any Other . . . • We can check the condition b 2 literally. � � b � � Considering the First . . . � Considering the First . . . • This means that we need to consider all possible values How Do We Check the . . . n 2 from 0 to b . How to Check the . . . • However, this can avoided if we: The Checking and Its . . . – multiply both sides of the inequality by b , and Home Page � � � 1 � < 1 � � Title Page – consider the equiv. inequality � b · 2 − n 2 b. � � � � ◭◭ ◮◮ • In this case, we can easily see that n 2 is the nearest ◭ ◮ � � � � 1 1 integer to the product b · 2: n 2 = b · . Page 8 of 11 2 Go Back • In these terms, the desired inequality takes the form Full Screen � � �� � � 1 1 � < 1 � � � b · 2 − b · b ; this is what we will check. � � Close 2 � � Quit
Outline Formulation of the . . . 8. How to Check the Third Condition Formulation of the . . . � � � 2 − n 3 1 � < 1 � � Are There Any Other . . . • We can check the condition 3 b 2 literally. � � b � � Considering the First . . . � Considering the First . . . • This means that we need to consider all possible values How Do We Check the . . . n 3 from 0 to b . How to Check the . . . • However, this can avoided if we: The Checking and Its . . . – multiply both sides of the inequality by b and Home Page � � � 1 � < 1 � � Title Page 3 – consider the equiv. inequality � b · 2 − n 3 b. � � � � ◭◭ ◮◮ • In this case, we can easily see that n 3 is the nearest ◭ ◮ � � � � 1 1 3 3 integer to the product b · 2: n 3 = b · . Page 9 of 11 2 Go Back • In these terms, the desired inequality takes the form Full Screen � � �� � � 1 1 � < 1 � � 3 3 2 − b · b ; this is what we will check. � � Close 2 � � � Quit
Outline Formulation of the . . . 9. The Checking and Its Results Formulation of the . . . • For each even number b from 2 to 10 9 , we checked Are There Any Other . . . whether this number satisfies both conditions Considering the First . . . � � �� � � �� Considering the First . . . � � � � 1 1 � < 1 1 1 � < 1 � � � � 3 3 � b · 2 − b · b ; 2 − b · b. � � � � How Do We Check the . . . 2 2 � � � � � How to Check the . . . • Result: for b ≤ 10 9 , both roots are only well approxi- The Checking and Its . . . Home Page mated for b = 2, b = 4, and b = 10. Title Page • Conclusion: only for these three bases, the desired con- ditions are satisfied. ◭◭ ◮◮ ◭ ◮ • This may explain why decimal and binary systems are the most frequently used. Page 10 of 11 • What we did: checked all the values b until 10 9 . Go Back • Conjecture: no other value b > 10 9 satisfies this ap- Full Screen proximation property. Close Quit
Recommend
More recommend
